Integral Sum Numbers of Graphs

The sum graph of a set \(S\) of positive integers is the graph \(G^+(S)\) having \(S\) as its vertex set, with two distinct vertices adjacent whenever their sum is in \(S\). If \(S\) is allowed to be a subset of all integers, the graph so obtained is called an integral sum graph. The integral sum number of a given graph \(G\) is the smallest number of isolated vertices which when added to \(G\) result in an integral sum graph. In this paper, we find the integral sum numbers of caterpillars, cycles, wheels, and complete bipartite graphs.